Central limit theorem
Objective. Standardise sums and assess normal approximations.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
The central limit theorem states that the standardised sum of many independent identically distributed variables with finite variance approaches a standard normal distribution, regardless of the original distribution's shape. For X1,...,Xn with mean μ and variance σ², the standardised sum Z = (X̄n − μ)/(σ/√n) converges in distribution to N(0,1) as n → ∞. This justifies approximating sums or averages of many small independent effects, such as measurement errors, by a normal distribution even when individual terms are far from normal.
Worked example
Using the central limit theorem, find the approximate standardised value of a sample mean X̄n with n = 100, population mean μ = 50, variance σ² = 25, when X̄n = 51.
- 1Identify σ = √25 = 5 and n = 100
- 2Compute the standard error σ/√n = 5/10 = 0.5
- 3Apply Z = (X̄n − μ)/(σ/√n) = (51−50)/0.5
- 4Simplify the standardised value
Answer: Z = 2
Concept mastery
Based on your past attempts at this lesson — the question types to practise again come first.
Complete a practice set to see your concept breakdown here.
Practice
Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.
Start practisingCommon mistake. Students often apply the normal approximation with too small a sample size or ignore the need for finite variance, assuming the theorem always applies exactly. Treat the result as an approximation that improves as n grows, not an exact statement for small n.
Your progress
Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.
0 of 5 steps complete
Your place saves automatically as you read.